User bernikov - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:39:51Z http://mathoverflow.net/feeds/user/8786 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106482/lambert-w-1x-as-x-rightarrow-0-asymptotic-behavior/106489#106489 Answer by Bernikov for Lambert $W_{-1}(x)$ as $x\rightarrow 0^-$: Asymptotic behavior Bernikov 2012-09-06T09:20:36Z 2012-09-06T09:20:36Z <p>First, if $x \in ]-1/e,0[$, one has $1 &lt; -W_{-1}(x) \leq -\frac{1}{x}$ (it is easy to prove that these inequalities are equivalent to $-1/e &lt; x \leq \frac{\exp(1/x)}{x}$, and that is true here). Then $0 &lt; \ln(-W_{-1}(x)) \leq -\ln(-x)$.</p> <p>Since $W_{-1}(x)\exp(W_{-1}(x)) = x$, the above inequalities yield $1 &lt; -W_{-1}(x) = -\ln(-x)+\ln(-W_{-1}(x)) \leq -2\ln(-x)$ (of course, one cas replace 2 by another suitable constant, if I only look for an asymptotic behavior, as we shall see). Finally, this gives $0 &lt; \ln(-W_{-1}(x)) &lt; 2\ln(-\ln(-x))$. Combine everything here, and you already have $W_{-1}(x) = \ln(-x) + O(\ln(-\ln(-x)))$ as $x \to 0^-$. One can do better, by reinjecting this estimate in the identity $W_{-1}(x) = \ln(-x) - \ln(-W_{-1}(x))$, and one obstains $W_{-1}(x) = \ln(-x) - \ln(-\ln(-x)) + O\left(\frac{\ln(-\ln(-x))}{\ln(-x)}\right)$.</p> <p>You may find some interets in reading "Asymptotic methods in analysis" by De Bruijn, the methods used to write $W_0$ as an infinite sum of powers involving $\ln$ can be used with $W_{-1}$, I guess.</p> http://mathoverflow.net/questions/24132/what-are-examples-of-mathematical-concepts-named-after-the-wrong-people-stigler/42167#42167 Answer by Bernikov for What are examples of mathematical concepts named after the wrong people? (Stigler's law) Bernikov 2010-10-14T16:18:43Z 2010-10-14T16:18:43Z <p>Liouville talked about the Legendre function when he studied the so-called Euler Gamma function. It made me doubt about who defined the Gamma function first.</p> http://mathoverflow.net/questions/37678/methods-for-additive-problems-in-number-theory Methods for "additive" problems in number theory Bernikov 2010-09-04T00:53:04Z 2010-09-04T06:38:03Z <p>I was wondering if there are some classical methods to tackle problems in number theory dealing with sums where the primes are not well-"controled". I talk about problems where we want to link a certain sum with information about the primes dividing the elements of the sum: the $abc$ conjecture is an example of such a problem, since we want to link $a$, $b$ and $a+b$ to the prime factors of $abc$, knowing that $a$, $b$ and $a+b$ are coprime. Another "additive" problem is the Goldbach conjecture.</p> <p>Since the natural way to deal with prime factors is for... factorization, these kinds of problems look way more complicated. Except sieve methods, are there any conclusive methods?</p> http://mathoverflow.net/questions/37128/about-a-non-obvious-link-between-the-jacobians-of-curves-and-differentials About a non-obvious (?) link between the jacobians of curves and differentials Bernikov 2010-08-30T08:19:33Z 2010-08-30T17:14:57Z <p>To explain my problem, I must give a lemma:</p> <blockquote> <p>Let $X$, $Y$, $Z$ be curves over $k$ (of characteristic 0) such that the genus of $Z$ is greater than 2, and $\pi : X \to Y$, $\phi : X \to Z$ two non-constant morphisms. If $\phi^\star(H^0(Z,\Omega))\subseteq\pi^\star(H^0(Y,\Omega))$, where $\Omega$ denotes the sheaf of regular 1-forms in each case, then there exists a non-constant morphism $u: Y \to Z$ such that $\phi = u \circ \pi$.</p> </blockquote> <p>Now, in a proof, I saw the use of this lemma, except that the hypothesis was the inclusion $\mathrm{Image}(\mathrm{Jac}(Z) \to \mathrm{Jac}(X)) \subseteq \mathrm{Image}(\mathrm{Jac}(Y) \to \mathrm{Jac}(X))$, instead of $\phi^\star(H^0(Z,\Omega))\subseteq\pi^\star(H^0(Y,\Omega))$. I can guess it is equivalent, but why? Is it related to Grothendieck's duality? Did I miss something obvious?</p> http://mathoverflow.net/questions/8825/how-many-different-representations-of-pi-can-we-come-up-with/36990#36990 Answer by Bernikov for How many different representations of pi can we come up with? Bernikov 2010-08-28T19:39:38Z 2010-08-28T19:39:38Z <p>I really like this formula: $1+\frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \frac{1}{1\cdot 3\cdot 5\cdot 7} + \frac{1}{1\cdot 3\cdot 5\cdot 7\cdot 9} + \cdots + {{1\over 1 + {1\over 1 + {2\over 1 + {3\over 1 + {4\over 1 + {5\over 1 + \cdots }}}}}}} = \sqrt{\frac{e\cdot\pi}{2}}$</p> http://mathoverflow.net/questions/36942/a-bound-for-the-manin-constant A bound for the Manin constant Bernikov 2010-08-27T23:52:01Z 2010-08-28T07:21:01Z <p>I recall that the Manin constant for a strong elliptic curve is a rational integer $c_E$ such that, for a modular parametrization $\phi: X_1(N) \to E$, one has $\phi^*(\omega_E)= 2\pi i c_E f(z)\mathrm{d}z$ ($f$ is the modular form associated to $E$). The Manin constant is supposed to equal +/-1, but it is not proved yet.</p> <p>I know that the prime divisors of $c_E$ are well known, but is there any bound for $c_E$, as the conducteur varies? If any, is there any reference for a proof? I heard that Edixhoven proved that $c_E$ is bounded (independently of $N$), but the article is forthcoming...</p> http://mathoverflow.net/questions/115988/does-the-linear-representations-of-an-finite-group-on-an-k-vector-space-forms-a-r Comment by Bernikov Bernikov 2012-12-10T15:51:30Z 2012-12-10T15:51:30Z Unless I miss something obvious, the elements of this set have no inverse for $\oplus$, hence it does not form a group, let alone a ring. http://mathoverflow.net/questions/109149/cyclotomic-polynomials-with-coefficients-0-pm1 Comment by Bernikov Bernikov 2012-10-09T09:35:59Z 2012-10-09T09:35:59Z My bad, I misread your question. http://mathoverflow.net/questions/109149/cyclotomic-polynomials-with-coefficients-0-pm1 Comment by Bernikov Bernikov 2012-10-09T09:29:17Z 2012-10-09T09:29:17Z It is not a characterization, but let $t$ be any positive integer, and take $p_1&lt; \dots &lt;p_t$ (these are $t$ prime numbers) such that $p_1 \geq 3$ and $p_1+p_2&gt;p_t$. Then the coefficients of $X^p$ and $X^{p−2}$ in $\Phi_{2p_1 \dots p_t}$ are $t-1$ and $t-2$ respectively. This way, one can provide an infinity of cyclotomic polynomials with the asked property. The details can be found in \emph{Polynomials} by Prasolov, there is a section about cylotomic polynomials. http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/36311#36311 Comment by Bernikov Bernikov 2012-07-17T07:31:39Z 2012-07-17T07:31:39Z Could you please give an example of such an isomorphism, or at least an argument to prove that these groups are isomorphic? I find it quite surprising. http://mathoverflow.net/questions/86990/effective-lower-bound-for-class-numbers-of-cyclotomic-fields Comment by Bernikov Bernikov 2012-02-08T12:51:41Z 2012-02-08T12:51:41Z I would be interested by the same question for imaginary quadratic fields. Is there anything known? http://mathoverflow.net/questions/87442/minimum-number-of-subsets Comment by Bernikov Bernikov 2012-02-03T15:58:57Z 2012-02-03T15:58:57Z Tell me if I am wrong, but it seems to be related to $(\nu,k,\lambda)$-designs, isn't it? http://mathoverflow.net/questions/86546/generating-function-related-to-2-residues-of-partitions Comment by Bernikov Bernikov 2012-01-25T11:04:32Z 2012-01-25T11:04:32Z $P(x)=\prod_{i=1}^\infty(1−x^i)^{−1}$ is the partition generating function. http://mathoverflow.net/questions/85236/examples-of-monster-groups Comment by Bernikov Bernikov 2012-01-09T13:38:06Z 2012-01-09T13:38:06Z The Pr&#252;fer p-group is an infinite group such that every propre subgroup is finite and cyclic (in fact, for every nonnegative integer n, there is a subgroup with $p^n$ elements). http://mathoverflow.net/questions/83229/quotients-of-perfect-powers-separated-by-an-integer Comment by Bernikov Bernikov 2011-12-12T10:08:45Z 2011-12-12T10:08:45Z Since the limit of $b(n)-a(n)$ is $\frac{1}{2}e &gt; 1$, this must holds for $n$ large too. http://mathoverflow.net/questions/70692/why-is-the-chebyshev-function-relevant-to-the-prime-number-theorem/70701#70701 Comment by Bernikov Bernikov 2011-07-21T02:23:37Z 2011-07-21T02:23:37Z I would say that the von Mangoldt function $\Lambda$ is more interesting than other multiplicative functions because of the relation $log = 1\star \Lambda$, where $\star$ is the convolution of arithmetic functions. This relation encodes the fundamental theorem of arithmetic, so &quot;heuristically&quot; it is of great interest, compared to the other propositions of the OP. http://mathoverflow.net/questions/63412/upper-bounds-for-the-sum-of-primes-n/63420#63420 Comment by Bernikov Bernikov 2011-04-29T15:11:40Z 2011-04-29T15:11:40Z Not exactly: there is a difference between the sum of primes less than or equal to $n$, and the sum of the $n$ first primes. I must have missed something: how could $s(n) \sim n^2 \ln(n)/2$ have not be proven before 1996 by Bach and Shallit? I do not see what is wrong with the fact that, since, $p_n \sim n \ln(n)$ (prime number theorem), one has: $\sum_{k=1}^n p_k \sim \sum_{k=1}^n k \ln(k) \sim \int_1^n t\ln(t) \mathrm{d}t \sim n^2\ln(n)/2$ http://mathoverflow.net/questions/37678/methods-for-additive-problems-in-number-theory Comment by Bernikov Bernikov 2010-09-04T07:22:47Z 2010-09-04T07:22:47Z @Scott: I did not look any text on additive number theory, and nothing really tough about combinatorics. @Gjergji: I just googled that theorem, it is nice! http://mathoverflow.net/questions/37128/about-a-non-obvious-link-between-the-jacobians-of-curves-and-differentials/37146#37146 Comment by Bernikov Bernikov 2010-08-31T02:49:18Z 2010-08-31T02:49:18Z What is the meaning of $T_0$? http://mathoverflow.net/questions/36942/a-bound-for-the-manin-constant/36951#36951 Comment by Bernikov Bernikov 2010-08-28T18:38:46Z 2010-08-28T18:38:46Z Your answer is all the more helpful, since the reference you quote learns to me that $p|c_E$ implies $p^2|4N$, and $4|c_E$ implies $4|N$. In the case I study, $E$ is given by $y^2=x(x−a)(x+b)$ where $a+b+c=0$ and $a, b, c$ are coprime. Thus $N$ is quadrafrei and $c_E$ is at most 2. Thanks (and I guess I should write to one of the authors for more details about the problem you raise)!